# Meson scattering amplitude in the linear sigma model

I am trying to calculate scattering amplitudes with linear sigma model Lagrangian, given as $$\mathcal L= \frac{1}{2}(\partial_{\mu}\sigma)^2+\frac{1}{2}(\partial_{\mu}\vec{\pi})^2-\mathcal U(\sigma,\vec{\pi})\tag{112}$$ Where, $$\mathcal U(\sigma,\vec{\pi})=\frac{\mathcal{\lambda}}{4}(\sigma^2+\vec \pi^2 -\mathcal f^2)^2-{H\sigma}.\tag{113}$$ the scattering amplitudes, for $$\pi\pi\rightarrow\pi\pi$$ and $$\sigma\sigma\rightarrow\sigma\sigma$$ are

$$\mathcal M_{{\pi}^a{\pi}^b;{\pi}^c{\pi}^d}=-2\lambda\Bigl(\frac{s-m_\pi^2}{s-m_\sigma^2}\delta_{ab}\delta_{cd}+\frac{t-m_\pi^2}{t-m_\sigma^2}\delta_{ac}\delta_{bd}+\frac{u-m_\pi^2}{u-m_\sigma^2}\delta_{ad}\delta_{bc}\Bigr)\tag{118}$$

$$\mathcal M_{{\sigma}{\sigma};{\sigma}{\sigma}}=-6\lambda-36\lambda^2\mathcal f_\pi^2\Bigl(\frac{1}{s-m_\sigma^2}+\frac{1}{t-m_\sigma^2}+\frac{1}{u-m_\sigma^2}\Bigr)\tag{121}$$

Where $$s$$,$$t$$ and $$u$$ are Mandelstam variables and $$\mathcal f_\pi^2=\frac{m_\sigma^2-m_\pi^2}{m_\sigma^2-3m_\pi^2}\mathcal f^2.\tag{117}$$ I do not know how to start, to get last two expressions (118) & (121), namely, $$\mathcal M_{{\pi}^a{\pi}^b;{\pi}^c{\pi}^d}$$ and $$\mathcal M_{{\sigma}{\sigma};{\sigma}{\sigma}}$$.

References:

• The expressions are written in the article arxiv.org/abs/arXiv:1006.0257 and i am following Peskin's book. – Suraj Kumar Rai Dec 27 '18 at 19:49
• How to get a one-loop scattering amplitude starting from a Lagrangian is basically the content of most textbooks on QFT, including Peskin's book. A very first step is rewriting your Lagrangian in terms of the vacuum expectation and fluctuation $v$ and $\Delta$ that appear in that paper. – octonion Dec 27 '18 at 22:36